Choose a method
Use the exponent as the number of equal growth periods. In a radical equation, isolate the radical before squaring and check the candidate afterward.
Avoid this trap
A 20% increase multiplies by 1.20, not 0.20. Rational expressions have excluded inputs wherever their original denominator equals zero.
Check and transfer
Compare values over equal intervals to distinguish a constant difference from a constant factor. Confirm that the exponent has the right time unit.
Worked example with a numerical check
A culture starts with 6 cells in a model and triples every hour. How many cells does the model predict after 3 hours? There are three complete growth periods. Multiply the initial count by three raised to the third power. The requested value is 162.
Contrast and diagnose
A machine worth $4000 loses 20% of its value each year in a model. What is its value after 2 years? A 20 percent decrease retains 80 percent each year. Apply the retention multiplier twice to the starting value. The requested value is 2560. An incorrect approach: Subtract the product of the loss rates rather than multiplying retention factors.
Transfer practice
For x ≠ 4, the expression (x² − 16)/(x − 4) equals x + c. What is c? Try first. Check: 4. Factor the numerator as a difference of squares. Cancel the shared factor only on the stated domain.
A culture starts with 6 cells in a model and triples every hour. How many cells does the model predict after 3 hours?
- There are three complete growth periods.
- Multiply the initial count by three raised to the third power.
- The requested value is 162.
Compare the other answer choices
Apply only one growth period.
Apply two growth periods instead of three.
Count the starting time as an extra growth period.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications